Fractional part ($\{x\}$)

The fractional part of a real number isIt is the amount left after removing the lower enclosing integer, even for negative inputs. Thus $\{-2.7\}=-2.7-(-3)=0.3$. It is not the signed string of digits obtained by truncation toward zero.

Translation by an integer leaves the fractional part unchanged: $\{x+m\}=\{x\}$. It is therefore a periodic function with period one, increasing from zero toward one on each unit interval, then returning to zero at the next integer. In Euclidean division $N=kq+r$ with $0\le r<k$, $\{N/k\}=r/k$; for $102/20$, this is $2/20=0.1$.

The notation is useful for following repeated motion modulo an integer. In normality and equidistribution under integer multiplication, the values $\{K^{n}x\}$ shift the digits of a positional notation; the fractional part discards the leading integer portion without losing the remaining digits. Its braces must be distinguished by context from the braces denoting a set.